I would think if he is interested in your food there is no harm in starting to offer some. Being able to sit unaided is a guide for all LOs starting solids but I would think if your health professional has advised you to begin then s/he must think your LO is ready (although if you don't feel ready for your LO to begin now I also see no harm in waiting a bit longer, mummy's instinct counts too). I used to put a folded towel behind DS in the high chair to help him sit, you might feel comfortable with that if he can hold his body and head up enough whilst eating? What you don't want is for him to be slumped or reclining at all as it's harder to move the food forward in his mouth.
It may also help to think of BLW more as 'leading' in the feeding rather than the usual idea of LO tucking into a vast array of foods just like Mummy and Daddy. If he is propped up well you could offer finger foods or a pre-loaded spoon with softer/mush/puree foods. If he doesn't have the muscle tone/development to lift the spoon or finger food unaided you can still allow him to lead the feeding by offering the spoon/food and letting him indicate he wants to eat it (open mouth, pulling your hand or the spoon towards him) and also following his cues for when to stop or the choice of which food he wants (closed mouth, turning away, not pulling food towards him, pushing food away, pushing food of the tray etc).
WRT intolerance/allergies you can still do BLW whilst trialing 1 food at a time and you could still eat together as a family which to me is a big part of the weaning journey even if LO doesn't eat much and doesn't stay for the entire meal.
